COVID Vaccine Transportation Access Act This bill establishes a grant program to remove transportation barriers in accessing COVID-19 vaccination sites. It also requires the federal government to cover the full cost of nonemergency transportation benefits related to accessing a COVID-19 vaccine for Medicaid recipients. Specifically, the Department of Health and Human Services must award the grants through state health agencies to nonprofits that have experience providing services to communities with disparities in health status and health care access. Grant-funded projects may include prescheduled or on-demand transportation to and from COVID-19 vaccination sites.